According to Kant, analytic judgements are made up of subject and a predicate. The predicate expresses nothing which isn't already contained in the subject.
Take this analytic judgement:
“All
bodies are extended.”
Here
the subject is “all bodies” and the predicate is “(are)
extended”. Within the subject is contained the concept [extended].
According to Kant, “I have not amplified the concept of body, but
only analysed it."
Now
take the following:
“All
bodies have weight.”
Here
the subject is again “all bodies” and this time the predicate is
“have weight”. This judgement is synthetic because the
predicate “have weight” isn't contained in the subject “all
bodies”. This predicate, therefore, “amplifies” rather than
“analyses” the subject.
We
now have two terms which sum up what's just been written:
“explicative
judgement” – adds nothing to the subject =
analytic
“ampliative
judgement” – increases the given cognition = synthetic
In
what way do we know the analytic judgement to be true or false? We
know it a priori. Take this judgement:
“Gold
is a yellow metal.”
The
concepts involved in the judgement above are empirical in
nature. However, the statement above, according to Kant, can
still be known to be true a priori. The reason for this is
that, again, the predicate “yellow metal” adds nothing to the
subject “gold”. According to Kant, we “require no experience
beyond our concept of gold”. Yellowness (as it were) is
contained in the concept [gold]. That means that the statement can be
known to be true a priori.
Arithmetic
According
to Kant, arithmetical judgements are all synthetic and not analytic,
as was commonly thought in his time (e.g. by Hume). However
7+
5 = 12
is still knowable a priori; though it's nevertheless synthetic. Arithmetical statements are a priori “because they carry with them necessity, which cannot be obtained from experience”.
Why
isn’t the above a mere analytic judgement such as “a = a”?
Why is it synthetic and a priori? After all
7
+ 5 = 12
basically
means
12
= 12
which
is a tautology of the kind
A
= A.
And
7
+ 5 = 13
would
be a contradiction of the kind
12
= 13
or
A
= B
Why
does Kant think that 7 + 5 = 12 is a priori as well as
synthetic? This is how Kant himself puts it:
“The
concept of twelve is by no means thought by merely thinking of the
combination of seven and five…”
Kant
continues:
“…analyse
this possible sum as we may, we shall not discover twelve in the
concept.”
We
can't find the concept [12] within the concept [7 + 5]. What more do
we need? According to Kant:
“We
must go beyond these concepts by calling to our aid some intuition
corresponding to one of them, i.e., either our five fingers or five
points…”
This
is very difficult to grasp without an explication of the notion of
intuition. However, it's the intuition itself that's
synthetic. Therefore the “five fingers or five points” needed for
the intuition are derived from experience. Therefore they're
synthetic (or the experience is). The judgement is synthetic a priori. To use Kant’s
terms, the concept [12] is an “amplification” of the concept [7 +
5].
What
about geometry?
Geometry
Take
the following principle of geometry:
“A
straight line is the shortest path between two points.”
According
to Kant, that statement is a synthetic judgement. Though it's also
knowable a priori. It's relatively easy to see why the above
is knowable a priori, but why is it also synthetic? Kant says:
“The
concept of the shortest is therefore altogether additional and cannot
be obtained by any analysis of the concept of the straight line.”
The
concept [shortest] isn't contained in the concept a [straight line]. Or, more accurately, the concept [the shortest path between
two points] isn't contained in the concept [a straight line]. Here
again, according to Kant, “intuition must come to aid us”.
Presumably here the (empirical) intuition is this:
a------------------------------------------b
That
is, a straight line between two points.
Kant
then summarises all the above. He calls synthetic a priori
judgements “apodeictic”; just as we would call an analytic
judgement “apodeictic”. Such judgements are apodeictic because
the predicate is already contained in the subject. However, unlike a
pure analytic judgement, such as
“All
bodies are bodies.”
we
need a “necessarily present intuition” which supplies the
synthetic part of the judgement or statement. Unlike the above
analytic statement,
“the
predicate [12] belongs to this concept [7 + 5] necessarily indeed,
yet not directly but indirectly by means of a necessarily present
intuition”.
Kant
went beyond the mere empirical synthesis of perceptions. He thought
that there is a priori synthesis too. When perceptions are
synthesized a priori, they are given, according to Kant,
“universal validity”.
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